Solve sqrt{8/3}divsqrt{1/45}

$$\sqrt{ \frac{ 8 }{ 3 } } \div \sqrt{ \frac{ 1 }{ 45 } }$$

Evaluate

$$2\sqrt{30}\approx 10.95445115$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{8}}{\sqrt{3}}}{\sqrt{\frac{1}{45}}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\frac{2\sqrt{2}}{\sqrt{3}}}{\sqrt{\frac{1}{45}}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{\frac{1}{45}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{2\sqrt{2}\sqrt{3}}{3}}{\sqrt{\frac{1}{45}}}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{2\sqrt{6}}{3}}{\sqrt{\frac{1}{45}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{45}} as the division of square roots \frac{\sqrt{1}}{\sqrt{45}}.

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{1}}{\sqrt{45}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{1}{\sqrt{45}}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{1}{3\sqrt{5}}}$$

Rationalize the denominator of \frac{1}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{5}}{3\times 5}}$$

Multiply 3 and 5 to get 15.

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{5}}{15}}$$

Divide \frac{2\sqrt{6}}{3} by \frac{\sqrt{5}}{15} by multiplying \frac{2\sqrt{6}}{3} by the reciprocal of \frac{\sqrt{5}}{15}.

$$\frac{2\sqrt{6}\times 15}{3\sqrt{5}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2\times 5\sqrt{6}}{\sqrt{5}}$$

Rationalize the denominator of \frac{2\times 5\sqrt{6}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{2\times 5\sqrt{6}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{2\times 5\sqrt{6}\sqrt{5}}{5}$$

Multiply 2 and 5 to get 10.

$$\frac{10\sqrt{6}\sqrt{5}}{5}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{10\sqrt{30}}{5}$$

Divide 10\sqrt{30} by 5 to get 2\sqrt{30}.

$$2\sqrt{30}$$